Document Open Access Logo

Twin-Width V: Linear Minors, Modular Counting, and Matrix Multiplication

Authors Édouard Bonnet , Ugo Giocanti, Patrice Ossona de Mendez , Stéphan Thomassé

Thumbnail PDF


  • Filesize: 0.88 MB
  • 16 pages

Document Identifiers

Author Details

Édouard Bonnet
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Ugo Giocanti
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Patrice Ossona de Mendez
  • Centre d'Analyse et de Mathématique Sociales CNRS UMR 8557, Paris, France
  • Computer Science Institute, Charles University (IUUK), Prague, Czech Republic
Stéphan Thomassé
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France

Cite AsGet BibTex

Édouard Bonnet, Ugo Giocanti, Patrice Ossona de Mendez, and Stéphan Thomassé. Twin-Width V: Linear Minors, Modular Counting, and Matrix Multiplication. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 15:1-15:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We continue developing the theory around the twin-width of totally ordered binary structures (or equivalently, matrices over a finite alphabet), initiated in the previous paper of the series. We first introduce the notion of parity and linear minors of a matrix, which consists of iteratively replacing consecutive rows or consecutive columns with a linear combination of them. We show that a matrix class (i.e., a set of matrices closed under taking submatrices) has bounded twin-width if and only if its linear-minor closure does not contain all matrices. We observe that the fixed-parameter tractable (FPT) algorithm for first-order model checking on structures given with an O(1)-sequence (certificate of bounded twin-width) and the fact that first-order transductions of bounded twin-width classes have bounded twin-width, both established in Twin-width I, extend to first-order logic with modular counting quantifiers. We make explicit a win-win argument obtained as a by-product of Twin-width IV, and somewhat similar to bidimensionality, that we call rank-bidimensionality. This generalizes the seminal work of Guillemot and Marx [SODA '14], which builds on the Marcus-Tardos theorem [JCTA '04]. It works on general matrices (not only on classes of bounded twin-width) and, for example, yields FPT algorithms deciding if a small matrix is a parity or a linear minor of another matrix given in input, or exactly computing the grid or mixed number of a given matrix (i.e., the maximum integer k such that the row set and the column set of the matrix can be partitioned into k intervals, with each of the k² defined cells containing a non-zero entry, or two distinct rows and two distinct columns, respectively). Armed with the above-mentioned extension to modular counting, we show that the twin-width of the product of two conformal matrices A, B (i.e., whose dimensions are such that AB is defined) over a finite field is bounded by a function of the twin-width of A, of B, and of the size of the field. Furthermore, if A and B are n × n matrices of twin-width d over F_q, we show that AB can be computed in time O_{d,q}(n² log n). We finally present an ad hoc algorithm to efficiently multiply two matrices of bounded twin-width, with a single-exponential dependence in the twin-width bound. More precisely, pipelined to observations and results of Pilipczuk et al. [STACS '22], we obtain the following. If the inputs are given in a compact tree-like form (witnessing twin-width at most d), called twin-decomposition of width d, then two n × n matrices A, B over F₂ can be multiplied in time 4^{d+o(d)}n, in the sense that a twin-decomposition of their product AB, with width 2^{d+o(d)}, is output within that time, and each entry of AB can be queried in time O_d(log log n). Furthermore, for every ε > 0, the query time can be brought to constant time O(1/ε) if the running time is increased to near-linear 4^{d+o(d)}n^{1+ε}. Notably, the running time is sublinear (essentially square root) in the number of (non-zero) entries.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Logic
  • Twin-width
  • matrices
  • parity and linear minors
  • model theory
  • linear algebra
  • matrix multiplication
  • algorithms
  • computational complexity


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. John T. Baldwin and Saharon Shelah. Second-order quantifiers and the complexity of theories. Notre Dame Journal of Formal Logic, 26(3):229-303, 1985. Google Scholar
  2. Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width II: small classes. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1977-1996, 2021. URL:
  3. Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width III: max independent set, min dominating set, and coloring. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 35:1-35:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL:
  4. Édouard Bonnet, Ugo Giocanti, Patrice Ossona de Mendez, Pierre Simon, Stéphan Thomassé, and Szymon Torunczyk. Twin-width IV: ordered graphs and matrices. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 924-937. ACM, 2022. URL:
  5. Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width I: tractable FO model checking. J. ACM, 69(1):3:1-3:46, 2022. URL:
  6. Édouard Bonnet, Jaroslav Nesetril, Patrice Ossona de Mendez, Sebastian Siebertz, and Stéphan Thomassé. Twin-width and permutations. CoRR, abs/2102.06880, 2021. URL:
  7. Steffen Börm, Lars Grasedyck, and Wolfgang Hackbusch. Introduction to hierarchical matrices with applications. Engineering analysis with boundary elements, 27(5):405-422, 2003. Google Scholar
  8. S. Braunfeld and M.C. Laskowski. Existential characterizations of monadic NIP, 2022. URL:
  9. Julia Chuzhoy and Zihan Tan. Towards tight(er) bounds for the excluded grid theorem. J. Comb. Theory, Ser. B, 146:219-265, 2021. URL:
  10. B. Courcelle. Graph rewriting: an algebraic and logic approach. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume 2, chapter 5, pages 142-193. Elsevier, Amsterdam, 1990. Google Scholar
  11. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125-150, 2000. URL:
  12. Marek Cygan, Fedor V Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized algorithms, volume 4. Springer, 2015. Google Scholar
  13. Sally Dong, Yin Tat Lee, and Guanghao Ye. A nearly-linear time algorithm for linear programs with small treewidth: a multiscale representation of robust central path. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1784-1797. ACM, 2021. URL:
  14. Rodney G. Downey, Michael R. Fellows, and Udayan Taylor. The parameterized complexity of relational database queries and an improved characterization of W[1]. In Douglas S. Bridges, Cristian S. Calude, Jeremy Gibbons, Steve Reeves, and Ian H. Witten, editors, First Conference of the Centre for Discrete Mathematics and Theoretical Computer Science, DMTCS 1996, Auckland, New Zealand, December, 9-13, 1996, pages 194-213. Springer-Verlag, Singapore, 1996. Google Scholar
  15. Yuli Eidelman and Israel Gohberg. On a new class of structured matrices. Integral Equations and Operator Theory, 34(3):293-324, 1999. Google Scholar
  16. Fedor V. Fomin, Erik D. Demaine, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. Bidimensionality. In Encyclopedia of Algorithms, pages 203-207. Springer, 2016. URL:
  17. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, Michal Pilipczuk, and Marcin Wrochna. Fully polynomial-time parameterized computations for graphs and matrices of low treewidth. ACM Trans. Algorithms, 14(3):34:1-34:45, 2018. URL:
  18. Jakub Gajarský, Michal Pilipczuk, Wojciech Przybyszewski, and Szymon Torunczyk. Twin-width and types. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 123:1-123:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL:
  19. Erich Grädel, Phokion G Kolaitis, Leonid Libkin, Maarten Marx, Joel Spencer, Moshe Y Vardi, Yde Venema, Scott Weinstein, et al. Finite Model Theory and its applications. Springer, 2007. Google Scholar
  20. Sylvain Guillemot and Dániel Marx. Finding small patterns in permutations in linear time. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 82-101, 2014. URL:
  21. Wolfgang Hackbusch. A sparse matrix arithmetic based on H-matrices. part I: introduction to h-matrices. Computing, 62(2):89-108, 1999. URL:
  22. Russell Impagliazzo and Ramamohan Paturi. On the Complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL:
  23. Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone. Handbook of Applied Cryptography. CRC Press, 2001. URL:
  24. Clément Pernet. Computing with quasiseparable matrices. In Sergei A. Abramov, Eugene V. Zima, and Xiao-Shan Gao, editors, Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC 2016, Waterloo, ON, Canada, July 19-22, 2016, pages 389-396. ACM, 2016. URL:
  25. Clément Pernet and Arne Storjohann. Time and space efficient generators for quasiseparable matrices. J. Symb. Comput., 85:224-246, 2018. URL:
  26. Michal Pilipczuk, Marek Sokolowski, and Anna Zych-Pawlewicz. Compact representation for matrices of bounded twin-width. In Petra Berenbrink and Benjamin Monmege, editors, 39th International Symposium on Theoretical Aspects of Computer Science, STACS 2022, March 15-18, 2022, Marseille, France (Virtual Conference), volume 219 of LIPIcs, pages 52:1-52:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL:
  27. Neil Robertson and Paul D. Seymour. Graph minors. I. Excluding a forest. J. Comb. Theory, Ser. B, 35(1):39-61, 1983. URL:
  28. Neil Robertson and Paul D. Seymour. Graph minors. V. Excluding a planar graph. Journal of Combinatorial Theory, Series B, 41(1):92-114, 1986. URL:
  29. Christopher De Sa, Albert Gu, Rohan Puttagunta, Christopher Ré, and Atri Rudra. A two-pronged progress in structured dense matrix vector multiplication. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1060-1079. SIAM, 2018. URL:
  30. Raf Vandebril, Marc Van Barel, Gene Golub, and Nicola Mastronardi. A bibliography on semiseparable matrices. Calcolo, 42(3):249-270, 2005. Google Scholar
  31. Jianlin Xia, Shivkumar Chandrasekaran, Ming Gu, and Xiaoye S. Li. Fast algorithms for hierarchically semiseparable matrices. Numer. Linear Algebra Appl., 17(6):953-976, 2010. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail